Interaction potential between heavy Q Q ¯ in color octet configuration in QGP
aa r X i v : . [ h e p - l a t ] S e p TIFR/TH/20-31
Interaction potential between heavy Q ¯ Q in color octet configuration in QGP Dibyendu Bala and Saumen Datta ∗ Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India.
We investigate the interaction between a heavy quark-antiquark pair in color octet configurationin gluon plasma. We calculate nonperturbatively an effective thermal potential for such a pairthrough the study of the correlation function of a hybrid state with Q ¯ Q octet and an adjoint gluonsource in the static limit. We discuss the extraction of an octet potential, and present results for theeffective thermal potential between octet Q ¯ Q pair in gluon plasma for moderately high temperatures . T c . The implications of our result are discussed. PACS numbers: 11.15.Ha, 12.38.Gc, 12.38.mh, 25.75.Nq
I. INTRODUCTION
Quarkonia, mesonic bound states of heavy quark and antiquark, provide one of the most important signatures ofthe formation of quark-gluon plasma (QGP) in relativistic heavy ion collision experiments. It was suggested in theearly days of collider studies of QGP [1] that the screening of the color charge inside QGP will lead to dissolution of
J/ψ states, which can be observed by modification of the dilepton peak. Various theoretical approaches have beenformulated to study the behavior of Q ¯ Q bound states in the plasma. See [2] for a recent comprehensive review [3]. Inparticular, an effective thermal potential approach to the problem has been formulated in [4] in perturbation theory,in the hard thermal loop (HTL) approximation. The behavior of such a potential has been examined in the effectivefield theory language for different distance scales [5]. While the formulation of Ref. [4] is for the theoretical abstractionof an isolated heavy Q ¯ Q placed in the plasma, the effective potential introduced there remains an important partof a description of heavy quark systems in plasma using the open quantum system language [6, 7]. It is possible toevaluate this potential nonperturbatively, from numerical calculations of thermal Wilson loop [8]. Calculations basedon Bayesian analysis [8, 9] or modelling of the low-frequency peak [10] suffer from large systematics. A new method,based on splitting of the thermal Wilson loop, has led to a well-controlled extraction of the potential from Wilsonloop data [11].For the phenomenology of quarkonia in plasma, one needs to know the interaction between the Q and the ¯ Q notonly in the singlet channel but also in the octet configuration. The interaction of the Q ¯ Q with the medium willchange its color configuration from singlet to octet and vice versa, and therefore study of evolution of the Q ¯ Q pair inplasma involves both color configurations. In particular, in the open quantum system approach developed to studyquarkonia in medium [6], the singlet and octet potentials are essential ingredients [7]. Also the Q ¯ Q pair may be ina color octet state at production time, and the time taken for it to go to a color singlet combination may be largerthan the formation time of the plasma. In particular, this is expected to be the case for quarkonia at large p ⊥ [12].However, nonperturbative information about the octet potential is not available in the literature. Part of the problemis the inherent difficulty in defining an octet state in a gauge-invariant set-up.For the singlet, the potential describes the time evolution of the thermal correlator of the nonrelativistic vectorcurrent, C > ( t, ~r ) ≡ Z d x (cid:10) χ † ( t, ~x ) σ k U † ψ ( t, ~x + ~r ) ψ † (0 , ~x + ~r ) σ k U χ (0 , ~x ) (cid:11) · (1)Here U is a suitable gauge connection such that the current is gauge invariant, ψ, χ are nonrelativistic fields thatannihilate a quark and create an antiquark, respectively, and the angular brackets denote thermal average. The σ k do not affect the O ( m Q ) potential. If one has a system where the sole interaction term is a potential V ( ~r ) betweenthe quark and the antiquark, then it is easy to show that, to leading order in 1 /M Q , C > ( t, ~r ) satisfies (cid:18) i ∂ t − ∇ ~r M Q (cid:19) C > ( t, ~r ) = V ( ~r ) C > ( t, ~r ) . (2) ∗ Electronic address: [email protected], [email protected]
We then define a potential [4] by equating the left hand side of Eq. (2) to V ( t, ~r ) C > ( t, ~r ) (staying within leading orderof 1 /M Q ), where the interaction effects are summarized in a time-dependent V ( t, ~r ). An effective thermal potential, V T ( ~r ), can then be defined in the large t limit, if the limit exists: V T ( ~r ) = lim t →∞ V ( t, ~r ). In the static limit, modulorenormalization factor, C > ( t, ~r ) reduces to a Minkowski-time Wilson loop: W M ( t, ~r ) = 13 Tr P e i R t dt A ( t ,~r ) U (cid:16) t ; ~r,~ (cid:17) P e i R t dt A ( t ,~ U (cid:16) ~ , ~r (cid:17) (3)and Eq. (2) reduces to i ∂ t log W M ( t, ~r ) −−−→ t →∞ V T ( ~r ) , (4)which defines the effective thermal potential for the singlet channel [4].Eq. (2) describes the time evolution of a thermal correlation function, and not of a Q ¯ Q wave function. It reducesnaturally to Eq. (4) in the static limit, which can lead to a nonperturbative calculation of the potential [8]. Theeffective thermal potential defined by Eq. (2) is complex in general [4]. It has been argued that the potential Eq.(2) can be re-interpreted in terms of evolution of a Q ¯ Q wave function [13, 14]. In this language, the evolution ofthe Q ¯ Q pair is described by a stochastic Hamiltonian, and V im T ( ~r ) is related to fluctuation of the stochastic noise.The effective thermal potential remains an essential ingredient in such open quantum system studies of quarkonia inplasma.For the octet potential, one can proceed in a similar way, starting with a point-split nonrelativistic current J a ( ~r ; ~x, ~x , t ) = ψ † ( ~x + ~r ; t ) σ k U ( ~x + ~r, ~x ; t ) T a U ( ~x , ~x ; t ) χ ( ~x ; t ) · (5)and looking at the time derivative of the correlator C JaJa > ( t, ~r ) a la Eq. (4). The current J a is gauge dependentand the correlator C JaJa > ( t, ~r ) needs to be defined in a fixed gauge. Unfortunately, standard gauge fixed definitionsof C JaJa > ( t, ~r ) lead to a system which may be very different from what was intended: e.g., in the temporal gauge C JaJa > ( t, ~r ) actually describes, in the static limit, a ¯ QQ a adj Q system [15]. In the literature one usually employs theCoulomb gauge; nonperturbatively, C JaJa > ( t, ~r ) is not defined in the Coulomb gauge, and a further fixing of temporalgauge along ~x gets us back to a ¯ QQ a adj Q system [15].When we talk about ¯ QQ in color octet combination in the context of quarkonia, we have in mind a system where¯ QQ is interacting with an adjoint gluonic source. To mimic this system, we could start with a trial current like J G ( ~r ; ~x, ~x , t ) = ¯ ψ ( ~x + ~r ; t ) U ( ~x + ~r, ~x ; t ) G a ( ~x ; t ) T a U ( ~x , ~x ; t ) χ ( ~x ; t ) , (6)which is a color singlet combination of the color-octet quark-antiquark system and an adjoint gluonic source at atime slice t , and then look at the correlator C G ( t, ~r ) = h J † G ( t ) J G (0) i . With a judicious choice of G , it is possibleto ensure that J G does not have overlap with a configuration where the quark-antiquark system is in color-singletstate. A color singlet state consisting of Q ¯ Q | o and adjoint gluon source is called a hybrid state. At zero temperature,hybrid potentials have been studied in detail in the literature [16]. For us, the important information is that in certainregimes, the hybrid current can be used to define an octet potential [17, 18].In this work, we study the thermal effect on the hybrid Wilson loop and extract information about the thermalmodification of the interaction potential between static Q ¯ Q in color octet configuration in gluon plasma. To ourknowledge, this is the first study of the effective thermal color octet potential, though a related quantity, the coloroctet free energy, has been studied before [19, 20]. Preliminary results of this study were presented in [21]. The cruxof the problem is to extract V T ( ~r ); this is discussed in Sec. II. We discuss our method in detail in Sec. II. Details ofthe numerical calculation are given in Sec. III. Our results for the potential are given in Sec. IV, and in Sec. V wesummarize and discuss the results. Some technical details are relegated to the appendix: Appendix A discusses roleof smearing in our study, some details relevant for Sec. II can be found in Appendix B, and various systematics ofour extraction of the potential can be found in Appendix C. II. OBSERVABLES AND ANALYSIS
For our study, we use the hybrid current operator Eq. (6) with the chromomagnetic field operator for G = G a T a :we use the two choices B z and B + = B x + iB y . Here z is taken to be the separation between the quark and theantiquark. In the static limit, one gets Wilson loop with insertion of the G field: W GM ( τ ; ~r, ~x ) = 13 Tr P e i R τ dτ A ( τ ,~r ) U ( τ ; ~r, ~x ) G † ( τ ; ~x ) U (cid:16) τ ; ~x,~ (cid:17) P e i R τ dτ A ( τ ,~ U (cid:16) ~ , ~x (cid:17) G (0; ~x ) U (0; ~x, ~r ) · (7)In order to define a potential, we will need to go to Minkowski time and take long time derivative, similar to Eq. (4).While the aim of the paper here is to calculate the effective thermal potential nonperturbatively, in order tounderstand the method, it helps to think in terms of perturbation theory. In leading order (LO) of perturbationtheory, the effect of the insertion G isolates and one gets the octet potential from the long time behavior of Wilsonloop: ≡ · · · (8)Here time direction is shown vertically, the grey circles indicate the magnetic fields, and the empty dots and dashedlines on the Wilson loop indicate color matrix insertion T a and 00 component of the gluon propagator, respectively.The leading order potential comes from a ladder sum of diagrams like those explicitly shown inside the parenthesesin Eq. (8), where the effect of the G insertion is merely a change in color factor due to the color matrix insertions inthe Wilson loops. Such factorization will not hold nonperturbatively, where J G for the B z operator will give rise topotential for the L=0 state Σ − u and the B + operator, that for the L=1 state Π u . Here L refers to angular momentumaround the axis joining the quark and antiquark, u denotes CP odd, and − refers to parity for reflection about aplane passing through this axis. The potential for these operators have been studied [16], and its connection to theoctet potential has been explored in detail [18]. In the deconfined phase, such hybrid states are not expected tosurvive. However, we will sometimes refer to the potentials obtained with the two operator insertions as L=0 andL=1 potentials, respectively.At short distances, we expect the potential for a state like Eq. (6) to give the potential for the color octet Q ¯ Q state,modulo a constant term capturing the effect of the G insertion [17, 18] : V G ( r ) ∼ V O ( r ) + Λ G + O ( r ) (9)Clearly, for the two operator choices here, Λ G is identical. So we expect V G to be same for the two operators in theshort distance regime, modulo O ( r ) effects. This behavior was tested in detail in [18]: while the convergence of V G extracted from the two components of the magnetic field was verified, it becomes difficult to isolate the color octetpotential due to the quick onset of the nonperturbative effects.At finite temperatures, we can define a thermal Wilson loop similar to Eq. (7), except the τ extent is now finite:0 < τ < β = 1 /T . Just as in the case of the singlet [4], one can define in HTL perturbation theory an effectivethermal potential by continuing to Minkowski time and taking a long time derivative a la Eq. (4). For completeness,we outline the steps in Appendix B. In this order, the factorization of Eq. (8) holds and we get, in the Debye screeningregime, a thermal potential V o ( ~r ; T ) = V o re ( ~r ; T ) − i V o im ( ~r ; T ) (10) V o re ( ~r ; T ) = g N c e − m D r πr − g C F π m D V o im ( ~r ; T ) = g T π (cid:20) N c Z ∞ dz z ( z + 1) − N c Z ∞ dz z ( z + 1) (cid:18) − sin zxzx (cid:19)(cid:21) where m D , the Debye mass, = gT in this order of perturbation theory (for gluon plasma), and x = m D r . It isinteresting to compare it to the thermal singlet potential [4]: V re T ( ~r ) = − g C F πr e − m D r − g C F π m D V im T ( ~r ) = g C F T π Z ∞ dz z ( z + 1) (cid:18) − sin zxzx (cid:19) · (11)Both V o im ( ~r ; T ) and V im T ( ~r ) → T C F g / π as r → ∞ . V re T ( ~r ) corresponds to the usual physics of Debye screening inmedium, such that for sufficiently large screening, the bound states will not form. On the other hand, V im T ( ~r ) clearlyleads to a broadening of the spectral function peak. It captures the physics of collision with the thermal particlesleading to a decoherence of the Q ¯ Q wave function [6, 14].The expressions Eq. (10) and Eq. (11) are valid only in the HTL limit T ≫ /r . For the distance regime rT ≪ a τ V r ee ff (r) τ /a τ W Ta W T τ −2.06−2.05−2.04−2.03−2.02−2.01−2.00−1.99−1.98 P ( τ ) FIG. 1: (Left) “Local mass” plot from W aT and W T for Set 3, at 1.5 T c , at r/a s = 6 (smearing level=200) for the Singlet.(Right) P ( τ ) = log W p T ( τ, ~r ), shown together with the value of the periodic part from Eq. (14) (red, full) and the first term ofEq. (15) (green, dashed). distances, V im T ( ~r ) ∼ r [5]. This is indeed the parametric behavior seen in the short distance regime nonperturbatively[11]. Eq. (10) is not expected to give us the correct potential, even qualitatively, at all distance scales; they, however,are compact and are useful in understanding certain features of the thermal potential.Our aim here is to extract the potential nonperturbatively from the Euclidean Wilson loop Eq. (7). For the singlet,nonperturbative assessment of V T ( ~r ) has been done, e.g., in [9, 10, 22], and recently in [11]. The potential, in particular V im T ( ~r ), is very different from Eq. (11) in the temperature range . T c . Here for the analysis of W G we will followthe strategy of [11], which we outline below. See [11] for a more detailed discussion.At zero temperature, modulo renormalization factors, the Minkowski space Wilson loop has the asymptotic timebehavior W M ∼ e − iV ( r ) t (Eq. (4)), leading to the Euclidean time behavior W E ∼ e − V ( r ) τ . Going to sufficiently long τ , this behavior is indicated by a plateau in − ∂ τ log W E , from which we extract V ( r ). At finite temperature, onedoes not see such a plateau behavior. It was pointed out in [11], however, that splitting the Wilson loop in parts“symmetric” and “asymmetric” around τ = β/ W a T ( τ, ~r ) = s W T ( τ, ~r ) W T ( β − τ, ~r ) , W p T ( τ, ~r ) = p W T ( τ, ~r ) × W T ( β − τ, ~r ) , (12)one can extract a plateau structure from log W a T ( τ, ~r ) ≈ ( β/ − τ ) V r (see also Sec. III). This behavior is illustratedin the left panel of Figure 1.The plateau from W a T ( τ, ~r ) gives the real part of the potential, while W p T ( τ, ~r ) contributes to the imaginary part.This can be understood by writing a spectral decomposition for P ( τ ) = log W p T ( τ, ~r ): P ( τ ) = Z ∞−∞ dω σ ( ω ; T ) 12 (cid:16) e − ωτ + e − ω ( β − τ ) (cid:17) + τ − independent terms τ → it ⇒ i∂ t P ( it ) = Z ∞−∞ dω σ ( ω ; T ) ω (cid:0) e − iωt − e − ωβ e iωt (cid:1) . (13)At large t , the oscillating factors exp( ± iωt ) ensure that only the ω → βω ) → P ( it ) leads to an imaginary potential.To proceed further, we note that exp( − i ω t ) − exp( i ω t − ω β ) t →∞ −−−→ − π i ω δ ( ω ). Then in order to get a finitepotential − i V im T ( ~r ) = lim t →∞ i ∂ t P ( it ) we need σ ( ω ; T ) ∼ ω → ω (1 + O ( ω )) . Since we expect thermal physics tointroduce a distribution function (1 + n B ( ω )) ω → −−−→ Tω in P ( τ ) (see Appendix B), the existence of a potential requiresa low ω structure of σ ( ω ; T ) ∼ (1 + n B ( ω )) β V im π ω . This leads to the following behavior of the Wilson loop near β/ W T ( τ, ~r ) = e − V re T ( ~r ) ( τ − β ) − βπ V im T ( ~r ) log sin ( π τβ ) − .... W T ( β/ , ~r ) (14)where the higher order terms .... are non-potential terms. For the periodic part, expanding σ ( ω ; T ) ((1 − exp( − βω ))in series of ω will give [11] ... = X l c l (2 l − β l (cid:18) ζ (cid:18) l, τβ (cid:19) + ζ (cid:18) l, − τβ (cid:19) − ζ (2 l, . (cid:19) (15)Just the simple form Eq. (14), without any corrections, gives a very good description of the Wilson loop data around β/
2. We show one illustration of this in Figure 1. Here we use Eq. (14) with the first term of Eq. (15) to fit thesinglet data. In the left plot we show the fitted value for V re T ( ~r ) on top of the ’local values’ obtained from W a T ( τ, ~r ).In the right panel we show P ( τ ), defined above Eq. (13), along with the contribution of the V im T ( ~r ) term in Eq. (14)and that of the c term. The V im T ( ~r ) term captures the behavior of the data near β/ χ );however, it can sometimes destabilize the plot if the interval is small, or if the data is not very accurate, as is oftenthe case for W G . Therefore for the octet, we stick to just the form in Eq. (14), and choose a suitable interval so thatthe fit quality is good. In Sec. IV we will show how well Eq. (14) explains the data, by examining plateaus for thelocal values of the potential.The discussion of the potential requires a low- ω peak. In order to successfully determine the potential, it is necessaryto get a region in τ where this peak dominates the contribution to the potential. In earlier literature, this peak hasbeen hypothesized to have a Lorentzian or Gaussian structure. However, the form in Eq. (14) leads to an asymmetricpeak: while for ω ≈ V r it gives a Breit-Wigner structure, the fall-off from the peak is very different in the large- ω andsmall- ω side. The limiting behaviors are ρ low ( r ; ω ) ≈ r π V im ( V re − ω ) + V | V re − ω | , V im ≪ T ∼ ( ω − V re ) − (cid:16) − βV im π (cid:17) ω − V re ≫ T (16) ∼ e − β ( V re − ω ) ( V re − ω ) − (cid:16) − βV im π (cid:17) ω − V re ≪ − T An asymmetric peak structure has also been suggested in Ref. [23], where the spectral function for thin Wilsonloops was discussed from HTL perturbation theory. It agrees with Eq. (16) near the peak, but starts disagreeing awayfrom it. The peak in Eq. (16) takes into account that the non-potential modes have been sufficiently suppressed bysmearing, so that only the potential part Eq. (14), contributes, as is supported by the data.
III. DETAILS OF THE CALCULATION
As mentioned in the previous section, our primary quantity is the nonperturbatively estimated value of the Wilsonloop W G , Eq. (7), in a gluonic plasma, at moderately high temperatures . T c . Since we need a very fine grid ofpoints for the extraction of the potential from the Wilson loop, we have used a space-time anisotropic discretization,with ξ = a s /a τ = 3. We have generated lattices with the anisotropic Wilson action S W = − V ( β s P s + β τ P t ) , where P s = 13 V N c X x X i 20 47 25 2002 2.57 16.53 38 32 × 25 91 100 150, 200,250,30032 × 32 91 100 150, 200, 250, 3003 2.60 16.98 45 40 × 23 91 90 150, 200, 250, 300, 40040 × 30 91 90 150, 200, 250, 300, 40040 × 38 89 90 150, 200, 250, 300, 40030 × 60 91 140 200 for set 1). The magnetic field operators have been implemented using the clover construction. As in the singlet case[11], we do APE smearing [26] of the spatial links to reduce the non-potential effects. At each APE step, a spatialgauge link is replaced by Proj SU (3) ( α × link + P spatial staples), where we kept α = 2.5. We have looked at datafrom a number of APE steps (shown in Table I). With higher number of APE steps, the data quality decreases, butthe effect of the non-potential terms also decreases, making it easier to reach a plateau and extract a potential.The use of spatial gauge link smearing is well-known in potential studies, and detailed nonperturbative studiesexist. In our context, it is instructive, however, to understand its effect in the leading order for the Wilson loop. thisis discussed in Appendix A. Some illustrations of its effect on the extracted potential can be found in Appendix C.In this work, we have given all physical quantities in temperature units. Conversion to physical units, if needed,can be done by setting T c to 280 MeV, which is the value obtained by fixing the string tension: √ σ = 0.44 GeV. Thespatial size of the lattices are > . IV. W G AND THE OCTET Q ¯ Q INTERACTION POTENTIAL Before presenting the results for the effective potential for C G ( t, ~r ), Eq. (6), we illustrate how well the form Eq.(14) explains the data, by doing the equivalent of a local mass plot: we extract the “local potential” from a subset ofdata points. We find V ( r ; τ ) from the data W ( r ; t ) with t = τ, τ + 1 , N t − τ, N t − τ − 1. If the data is dominated bythe potential term near β/ 2, we will expect a plateau near β/ 2. We give two examples of such an effective potentialplateau in Figure 2. To get the results, we have done a bootstrap analysis, where the parameter values within eachbootstrap sample are obtained by a χ fit with the full covariance matrix. The data is seen to show a plateau behaviorin a region around β/ 2. The final value of the potential (within each bootstrap sample) is then obtained by doinga fit to Eq. (14) over this plateau range. The statistical error for a given fit range is the (16,84) percentile band ofthe bootstrap distribution. The quoted errors in Sec. IV A and Sec. IV B also include effect of varying the fit rangewithin the plateau region, and spread over smearing levels (see Sec. C).We present results for the real part of the potential in Sec. IV A, and in Sec. IV B the results for the imaginarypart are shown. Discussion of various systematics related to the results presented in this section have been put inAppendix C. A. V o re ( ~r ; T ) As discussed in Sec. II, the results for V o re ( ~r ; T ) are obtained from W a T ( τ, ~r ) with the hybrid current operators. At T = 0 the hybrid potential has been studied in detail in the literature. In Figure 3 we show the potential obtained byus below T c for the two operator insertions B z and B + . Strictly speaking the lattices here are at 0.75 T c (see TableI); but in gluon plasma one expects very little temperature effect at this temperature, and we indeed checked that ourresults are in very good agreement with a recent analysis of T=0 hybrid potential, Ref. [16]. Since this is, in effect,a zero temperature potential, we obtained the potential from a conventional exponential fit.From Eq. (6) we would expect that, at short distance, the potentials extracted for the two channels would agreeand give the octet potential, modulo an additive constant. As the figure shows, the potentials do seem to agree atvery short distances . . z , 1.2 T c r/a s =10 a t V r e a t V i m t/a t + , 1.2 T c r/a s =12 a t V r e a t V i m t/a t FIG. 2: “Local values” of the potential, V o re ( ~r ; T ) and V im T ( ~r ), for Set 3, at 1.2 T c . Results for smearing level 300 are shown.The results quoted in Sec. IV A and Sec. IV B, obtained from fits over the plateau range, are shown with horizontal bands.The filled diamonds and dotted lines show results for V im T ( ~r ) while the empty circles and dashed lines show results for V re T ( ~r ).(Left panel) results for L=0, at r = 10 a s . (Right) Those for L=1, at r = 12 a s . V r e (r) / T c rT c B z B + FIG. 3: Hybrid potentials below T c for the L=0 ( B z ) and L=1 ( B + ) channels. The results are obtained from the 30 × attractive at long distances, supporting bound states for Σ − u and Π u respectively, quantitatively the long distanceattractive part is very different for the two channels, and also from the long distance part of the singlet.At finite temperatures, the long-distance nonperturbative behavior is suppressed, and one may expect to be ableto extract information about the octet potential over longer distances. This is exactly what we found for V o re ( ~r ; T ).We extract the potential from W a T ( τ, ~r ), as explained in Sec. II.In Figure 4 we compare the results of V o re ( ~r ; T ) extracted from the Wilson loop with B z and B + insertions. Notethat the potential can be extracted from the Wilson loop modulo an additive renormalization constant (see AppendixB). For the results in this section, we have fixed the additive renormalization constant by demanding that the T=0singlet potential at the shortest distance r = a s matches the lattice discretized Coulomb potential: V s ( r = a s , T = 0) = − g C F Z d k π cos k a s P i sin ( k i a s / 2) (18)where for the coupling g we have used the “boosted coupling” g ( r ∼ a ) = 6 √ β s β τ √ P s P t ; P s , P t are the plaquettevariables defined in Eq. (17). The choice of the coupling and the matching point is somewhat arbitrary, and nodetailed study of systematics of the subtraction was done; so the potentials should be taken to be defined modulo asmall, temperature-independent additive constant. We stress that the additive normalization of the singlet at T = 0fixes the renormalization both V s re ( ~r ; T ) and V o re ( ~r ; T ) at all temperatures.The difference between Figure 4 and Figure 3 is stark: there is no nonperturbative rising part of the potentialabove T c , and the potentials for L=0 and L=1 agree very well (within errors) to the distance studied. Here, therefore,free from any dominant effects of the gluon string, one can talk about a “octet” potential, which is related to theinteraction between the heavy quark and antiquark in the color octet configuration. -2.2-2-1.8-1.6-1.4 0 0.4 0.8 1.21.2 T c V r e (r) / T c rT c B z B + -2.8-2.6-2.4-2.2-2 0 0.4 0.8 1.21.5 T c V r e (r) / T c rT c B z B + -3-2.8-2.6-2.4 0 0.4 0.8 1.22.0 T c V r e (r) / T c rT c B z B + FIG. 4: Comparison of the V o re ( ~r ; T ) obtained for the L=0 ( B z ) and L=1 ( B + ) channels. (Left) 1.2 T c , (middle) 1.5 T c and(right) 2.0 T c . The points for B + have been shifted slightly along x-axis in the plot. -4-3-2-1 0 1 0 0.4 0.8 1.2 1.6B z V r e (r) / T c rT c c - 21.2 T c c c -4-3-2-1 0 0 0.4 0.8 1.2 1.6B + V r e (r) / T c rT c c - 21.2 T c c c FIG. 5: Temperature dependence of V o re ( ~r ; T ); (left) L=0 and (right) L=1 channels. As indicated in the label, a constant hasbeen subtracted from the potential below T c for convenience of showing it with the above T c results. In Figure 5 we show V o re ( ~r ; T ) at different temperatures above T c . For comparison, the hybrid potential of Figure3 is also shown in the same plot. A constant has been subtracted from the hybrid potential below T c for showing itin the same scale. This figure clearly displays the effect of the deconfinement transition on the potential: the octetpotential above T c is repulsive at all distances. There is no trace of the long distance nonperturbative attractive partpresent in the potential of the hybrid operator below T c .While this is the first nonperturbative study of V o re ( ~r ; T ) from lattice, a related quantity, the free energy of a coloroctet Q ¯ Q pair in the plasma, has been nonperturbatively studied before [19, 20]. Unlike the thermal potential, it isstraightforward to nonperturbatively define the color octet free energy of a Q ¯ Q pair in the Coulomb gauge. The freeenergy is real, and is identical to the potential at T=0. In the plasma, it has been found to be close to the real partof the potential; see [11] for a comparison of the two for the singlet channel. The color octet free energy defined inCoulomb gauge shows screening, and is qualitatively similar to the behavior of V o re ( ~r ; T ) shown in Figure 5. A detailedanalysis of the short distance behavior of the Coulomb gauge fixed color octet free energy has been done in Ref. [20],and has been found to be in excellent agreement with perturbation theory.In Figure 6 we display together the octet and singlet potentials above T c . The octet potential is much flatterthan the singlet potential. Also at each temperature, the two potentials approach the same temperature-dependentconstant. Physically one expects this; at sufficiently long distance the interaction between the Q and the ¯ Q isexpected to vanish; the remnant constant then may be interpreted as a thermal correction to mass of the quark. Inthe right panel of Figure 6 we check this behavior down to longer distances using the larger, but coarser, set 1 data.Both these behaviors are qualitatively consistent with the expectations from perturbation theory, Eq. (10) and Eq.(11). We also note that the convergence of the singlet and octet potentials happen at shorter distances at highertemperatures. This is also expected, since the difference is ∝ e − m D r r , which is smaller at higher temperatures, where m D is larger. However, there are some quantitative differences from perturbation theory: in particular, the differencesin the asymptotic values at two temperatures is larger than what Eq. (10) predicts.To further check the conformity with the perturbative behavior, we look at δ V re T ( r ) = V re T ( ~r )( r + 1) − V re T ( ~r )( r ).In leading order perturbation theory, the ratio of this quantity in singlet and octet channels is − ( N c − -6-5-4-3-2-1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.61.2 T c c c V r e (r) / T c rT c singletB z B + -3.6-3-2.4 0 0.5 1 1.5 2 2.5B + c Singlet V r e (r) / T c rT c a t =1/30T c a t =1/45T c FIG. 6: (Left) Comparison of the singlet and octet potentials V re T ( ~r ) above T c . For V o re ( ~r ; T ), results for both L=0 and L=1channels are shown (with the L=1 points slightly shifted along x axis for ease of viewing). In the right panel, we show thiscomparison down to a distance rT c ∼ . 5, for 1.5 T c . In this plot the long distance part is from the coarser lattice of set 1. -1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 1.41.2 T c ( δ V s r e , - δ V o r e ) / T c rT c B z B + singlet -1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 1.41.5 T c ( δ V s r e , - δ V o r e ) / T c rT c B z B + singlet -1 0 1 2 0 0.2 0.4 0.6 0.8 1 1.2 1.42.0 T c ( δ V s r e , - δ V o r e ) / T c rT c B z B + singlet FIG. 7: Comparison of δ V re T ( r ) = V re T ( ~r )( r + 1) − V re T ( ~r )( r ) for singlet and octet channels, at 1.2 T c (left), 1.5 T c (middle) and2 T c (right). The points for B z have been slightly shifted along the horizontal axis. B. V o im ( ~r ; T ) The effective thermal potential is in general complex [4], with the imaginary part of the potential related to dampingand decoherence mechanisms [6]. In the case of singlet channel, where the potential is attractive and leads to spectralfunction peaks for sufficiently massive quarks, the imaginary part controls the width of the spectral function peak.Such an interpretation is not available here, and the imaginary part is to be understood as introducing decoherencein the Q ¯ Q system during its evolution as octet.The extraction of the imaginary part from the hybrid operators of the sort used here turns out to be more prob-lematic, and we can only get partial information about them. In particular, we were not able to get reliable resultsfor V o im ( ~r ; T ) at 2 T c , and show results only for 1.2 T c and 1.5 T c .In Figure 8 we compare the results for V o im ( ~r ; T ) with insertion of the B z and B + operators. Here we see a differentbehavior from that seen in Figure 4 for V o re ( ~r ; T ): while at short distances, the results for the two insertions agreewithin statistical error, at longer distances rT & V o im ( ~r ; T ) as color octet potential, as it has contributions from the gluonicoperator insertion.In Figure 9 we show the temperature dependence of V o im ( ~r ; T ). Below T c V o im ( ~r ; T ) is consistent with zero. Above T c it is very different from zero. In particular, the most striking behavior of V o im ( ~r ; T ) is that at r → V s im ( ~r ; T )( r → → V s im ( ~r ; T ) vanishes at r → 0, but rises ∼ r c V i m (r) / T rTsingletB z B + c V i m (r) / T rTsingletB z B + FIG. 8: V o im ( ~r ; T ) extracted from L=0 ( B z ) and L=1 ( B + ) channels, at 1.2 T c (left) and 1.5 T c (right). The points for B + havebeen slightly shifted horizontally. The singlet channel result at the same temperature, V s im ( ~r ; T ), is also shown. z V i m (r) / T c rT c c c + V i m (r) / T c rT c c c FIG. 9: Temperature dependence of V o im ( ~r ; T ), for L=0 (left) and L=1 (right) channels. The points for 1.5 T c have been slightlyshifted along the horizontal axis. at small r [11]. The slope of V o im ( ~r ; T ) at small r is smaller than that of V s im ( ~r ; T ). Within the accuracy of our results, V o im ( ~r ; T ) for rT < V o im ( ~r ; T ) decreasing with r ,as suggested by Eq. (10). V. DISCUSSION One way to understand the behavior of a heavy Q ¯ Q pair in quark-gluon plasma is through the introduction of aneffective thermal potential [4, 5, 14]. In order to understand the evolution of quarkonia in plasma, we need to knowthe effective potential of Q ¯ Q pair in both singlet and octet color configurations.Nonperturbative information about the singlet potential is available in the literature [11]. The real part of thepotential shows the expected medium screening, but at a quantitative level, differs from the leading order perturbativepotential even at 2 T c . The deviation from perturbation theory is even stronger in the imaginary part of the potential.In contrast to the singlet potential, very little is known nonperturbatively about the in-medium interaction of Q ¯ Q in a color octet configuration. One reason for this is the difficulty in nonperturbatively defining a color octet potential.In this paper, we have made the first nonperturbative study of the effective interaction potential for Q ¯ Q in color octetconfiguration in the plasma. The color-octet state is studied by looking at gauge invariant states formed by combininggluonic operators with color octet static Q ¯ Q source. At T = 0 potentials for such states, called hybrid states, havebeen studied in detail: in the perturbative regime at small r , they are expected to give information about the octetpotential, while at longer distances, where nonperturbative effects dominate, the potential becomes dependent ondetails of the gluonic operator.In contrast, we find that the in-medium potential V o re ( ~r ; T ) above T c remains independent of the specific hybridchannel, and gives information about the interaction between the octet Q ¯ Q pair. This is illustrated in Figure 4. Our1results for the color octet potential is summarized in Figure 5. The color octet potential is found to be screenedabove T c , and is repulsive at all distances. This indicates that there will not be any bound states of the heavy Q ¯ Q in the plasma in the color octet configuration. At long distances, the potential agrees with the color singlet potential(Figure 6). Within the accuracy of our data, the data is also consistent with the leading order scaling behavior δV o re ( ~r ; T ) δV s re ( ~r ; T ) = − C A / − C F C F ; this is demonstrated in Figure 7.The thermal effective potential is known to have an imaginary part [4]. The imaginary part is related to the physicsof Landau damping and decoherence of the wave function of the Q ¯ Q state. The effective potential obtained from thehybrid state also has an imaginary component. The extraction of this part is more difficult, and our results for V im T ( ~r )have large errors. The imaginary part of the extracted potential is more difficult to interpret in terms of a color octetpotential. For the two hybrid operators we looked at, we found agreement in V o im ( ~r ; T ) only up to distances rT ∼ V s im ( ~r ; T ) approaches zero at short distances. On the other hand, V o im ( ~r ; T ) acquiresa nonzero value even at r → T c . This is consistent with the behavior predicted in perturbation theory,and is also in line with physical intuition [6]. The singlet Q ¯ Q at very short distances will look like a colorless objectto the medium particles, which will not be able to resolve its structure. On the other hand, the medium particleswill interact strongly with the color octet Q ¯ Q , leading to damping. The r dependence of the imaginary part is muchmilder than that of the singlet in the region rT . 1; within the (limited) accuracy of our calculation, V o im ( ~r ; T ) isconsistent with a constant in this region. Acknowledgements: We would like to thank Gunnar Bali, Nora Brambilla, Peter Petreczky, Anurag Tiwari andAntonio Vaio for discussions. This work was carried out under the umbrella of ILGTI. The computations reportedhere were performed on the clusters of the Department of Theoretical Physics, TIFR. We would like to thank AjaySalve and Kapil Ghadiali for technical support. Appendix A: APE smearing As discussed in Sec. III, the extraction of the potential from thin link Wilson loops is difficult, and we do APEsmearing [26] of the spatial gauge links. This consists of replacing the spatial gauge links U i ( ~x, τ ) → Proj SU (3) h α U i ( ~x, τ ) + X ≤ j ≤ j = i n U j ( ~x, τ ) U i ( ~x + a s ˆ j, τ ) U † j ( ~x + a s ˆ i, τ ) (A1)+ U † j ( ~x − a s ˆ j, τ ) U i ( ~x − a s ˆ j, τ ) U j ( ~x − a s ˆ j + a s ˆ i, τ ) oi iteratively. While the quality of the signal for the Wilson loop detoriates with the number of smearing steps, theeffect of non-potential terms also decrease, making extraction of potential easier. For this work, we have taken α =2.5, and have done up to 400 steps of APE smearing.It is instructive to see the effect of APE smearing on the leading order expressions for the Wilson loops. Following[27] we write the effect of smearing on the gauge fields, A µ , where V ( x, x + aµ ) = ei a A µ ( x ). To linear order, A Ni ( Q ) = n f N (ˆ ~q ) P Tij (ˆ q ) + P Lij (ˆ q ) o A j ( Q ) (A2)where f (ˆ q ) = (1 − c q ) ∼ e − c ˆ q , ˆ q = X i =1 ˆ q i , ˆ q i = 2 sin q i a s / ,c = 44 + α , and the projection operators are defined above. For small q i a s , ˆ q i → q i a s and the projection operatorsbecome P T,Lij ( q ) .Then the propagator of the smeared fields, G Nij ( Q ) = h A Ni ( Q ) A Nj ( − Q ) i ∼ f N ( q ) P Tij ( q ) Q + Π T ( ω q , ~q ) (A3)2leads to the spectral function representation G Nij ( q ) ≡ Z ∞−∞ dq π ˜ ρ T ( q , ~q ) q − iω k , ˜ ρ T ( q , ~q ) ∼ f N ( q ) ρ T ( q , ~q ) · (A4)This results in a suppression of the nonpotential contribution to Wilson loop, as we discuss in the next section. Appendix B: LO calculation of potential in HTL Various strategies in our nonperturbative calculation of the potential has been motivated by insights from pertur-bation theory and in particular, the expression for the Wilson loop in LO HTL approximation. Here we put togetherthe leading order results for the thin Wilson loop, 3 and 7, in this approximation. This section follows Ref. [4].We use the Coulomb gauge. Then the gluon propagators are: D ( ω n , ~k ) = 1 K + Π E ( K ) K ~k , D ij ( ω n , ~k ) = 1 K + Π T ( K ) (cid:18) δ ij − k i k j ~k (cid:19) · (B1)Here K refers to the Euclidean four-momenta ( ω n , ~k ). The spectral functions ρ E ( k , ~k ) , ρ T ( k , ~k ), introduced throughthe integral relations 1 K + Π T,E ( K ) = Z ∞−∞ dk π ρ T,E ( k , ~k ) k − iω k , (B2)provide the connection to Minkowski momenta.For the singlet channel, the potential in LO will come from diagrams for ordinary Wilson loop similar to the onesshown within parentheses of Eq. (8). They add up to g C F Z d k π (cos k r − ( τ~k + Π E (0 , ~k ) + Z ∞−∞ dk π ρ E ( k , ~k ) (1 + n B ( k )) (cid:18) ~k − k (cid:19) (1 + e − βk − F ( k , τ )) ) (B3)where we define the symmetric and antisymmetric functions F ( k , τ ) = e − k τ + e − ( β − τ ) k , G ( k , τ ) = e − k τ − e − ( β − τ ) k . (B4)The term linear in τ in Eq. (B3) survives in W a T ( τ, ~r ), leading to the potential V r ≡ g C F Z d k π cos k r − ~k + Π E (0 , ~k ) = − g C F πr e − m D r + g C F (cid:16) − m D π + I a (cid:17) (B5)where the additive divergent term I a = R d k π ~k ∼ a in Eq. (B5) results from defining the potential through Wilsonloop, forcing V r ( r → → 0. The standard convention of defining potential, used in Eq. (11), sets I a → 0, so thatwe get the familiar Coulomb potential at short distances. This has been done in Sec. IV by fixing the T=0 singletpotential at r = a s through Eq. (18).Going to Minskowski time and taking large t , using the relationlim t →∞ i∂ t F ( k , τ ) | τ → it = lim t →∞ k (cid:0) e − ik t − e − βk e ik t (cid:1) → − k πiδ ( k ) · (B6)we see that the potential picks up contribution from ρ E ( k → , ~k → | k | ≪ | ~k | , ρ T ( k , ~k ) , ρ E ( k , ~k ) in Eq. (B2) behave like [4] ρ E ( k , ~k ) ≈ − πm D k | ~k | ( k + m D ) , ρ T ( k , ~k ) ≈ πm D ω | ~q | . (B7)The term with 1 /k in the second term of Eq. (B3) then leads to V im T ( ~r ) in Eq. (11). The 1 /k term does not leadto a potential; Figure 1 indicates that the contribution of this term is small near τ = β/ 2. As discussed in Sec. A,smearing will suppress ρ T ( k , ~k ).3The diagrams (B8)add up to g C F Z d k π (cos k r − (cid:18) k − ~k (cid:19) Z ∞−∞ dk π ρ T ( k , ~k ) (1 + n B ( k )) (1 + e − βk − F ( k , τ )) . (B9)Here the gluon lines correspond to transverse gluon propagators. Eq. (B9) does not contribute to the potential, as canbe seen using Eq. (B6). However, they will contribute to the fit near τ ∼ β/ 2. These terms, however, have ρ T ( k , ~k );as explained in Sec. A, smearing leads to a strong suppression of these terms. When the results for potential stabilizewith number of smearing steps, it indicates that the contribution of these terms have become negligible and we aregetting contribution from the potential terms only.The discussion for the hybrid Wilson loop is similar. The potential contributions in LO come from the diagramsexplicitly shown in the rhs of Eq. (8), summing up to h BB i × ( Z d k π (cid:18) g C F + g N c e ik r (cid:19) " − τ~k + Π E (0 , ~k ) (B10)+ Z ∞−∞ dk π ρ E ( k , ~k ) (cid:18) k − ~k (cid:19) (1 + n B ( k )) (cid:0) e − βk − F ( k , τ ) (cid:1)(cid:21)(cid:27) Renormalizing in the same way as the singlet leads to the potentials Eq. (10). We reiterate that the additiverenormalization we have used is fixed by matching of the T=0 singlet potential at r/a s = 1: no separate additiverenormalization is used for the octet.Other diagrams included in · · · in Eq. (8) are (B11)and variations: where the gluon lines are at τ = 0 or to the right of B , etc. The sum of their contributions is h BB i × (cid:26) g N c Z d k π − cos k rk − g N c Z d k π − cos k x − cos k ( r − x ) k (cid:27)Z ∞−∞ dk π ρ T ( k , ~k ) (1 + n B ( k )) (cid:0) e − βk − F ( k, τ ) (cid:1) · (B12)Using Eq. (B6) we see that they do not contribute to the potential. In the Euclidean time data, smearing suppressestheir contribution, due to the ρ T ( k , ~k ) terms.Diagrams that do not satisfy the factorization behavior of Eq. (8) are+ Variations + + Variation (B13)The left set involves only the symmetric function F ( k, τ ) whereas the right set involves both F ( k, τ ) and G ( k, τ ) ofEq. (B4). The expressions are straightforward, if unilluminating; it is easy to check that they do not contribute tothe potential. Also they both involve two or more factors of ρ T ( k , ~k ) and can be suppressed by smearing.4 z c a τ V r e (r) r/a s + c a τ V r e (r) r/a s z c a τ V r e (r) r/a s + c a τ V r e (r) r/a s smr 150smr 250smr 300smr 400 FIG. 10: Smearing dependence of our V o re ( ~r ; T ) extracted from the smeared Wilson loops W G . The top row shows results at 1.2 T c while the bottom row shows results at 1.5 T c . The panels to the left are for B z and those to the right are for B + insertions.For ease of viewing, some sets have been slightly shifted along x axis in the plot. The diagrams Eq. (B12) and Eq. (B13) do not contribute to the potential; however, they contribute in the finite τ Euclidean Wilson loops. For a successful extraction of the potential, we need to identify a plateau where theircontributions are negligible. As the structures of these terms demonstrate, smearing lead to their suppression; that iswhy we get the plateaus demonstrated in Figure 2, from where we can extract the potential. Appendix C: Systematics in potential estimation In this section we discuss the effect of smearing on our extracted potential, and the size of the discretization errorin our results. V o re ( ~r ; T ) As discussed in the text, for the spatial gauge connections U in Eq. (7) we have used APE smeared links. Smearingalso affects the G field. For the singlet potential, it was noticed that the extraction of the potential depends onthe smearing level to some extent [11]: one gets a better identified plateau, and the extracted potential seems tochange with the smearing level at small levels of smearing, before stabilising at some level. A similar trend is seen inthe octet case, except the effect is somewhat enhanced, and one needs to go to higher levels of smearing before theresult becomes insensitive to the smearing level (within our errors). At higher levels of smearing, a better plateau isobtained; at the same time the statistical noise increases. In Figure 10 we show the potential extracted from Wilsonloops with different levels of APE smearing, for set 3. We find that the potential saturates only at 300 smearing stepsat this cutoff. For the results quoted, we have included a systematic error covering the spread between results from300 and 400 levels of smearing. For comparison, for the same set, the singlet potential stabilised by 200 smearingsteps. For set 2, we find that 200 smearing steps is enough to stabilise the potential, and the sytematic error coversdata with 200 and 250 smearing steps. For Figure 10 as well as for other figures shown in cutoff units, we show theunrenormalized data (i.e., the matching to Eq. (18) is not done).The lattice-discretized results will have discretization errors, which go to 0 as one takes the continuum limit. Ourlattices are quite fine-grained, so discretization effects are expected to be small. In order to estimate the size of thediscretization error, in Figure 11 we show the results for V o re ( ~r ; T ) extracted from two different lattice spacings. Here5 -2.2-2-1.8-1.6-1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4B z , 1.2T c V r e (r) / T c rT c a τ =1/38T c a τ =1/45T c -2.2-2-1.8-1.6-1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4B + , 1.2T c V r e (r) / T c rT c a τ =1/38T c a τ =1/45T c -2.6-2.4-2.2-2 0 0.2 0.4 0.6 0.8 1 1.2 1.4B z , 1.5T c V r e (r) / T c rT c a τ =1/38T c a τ =1/45T c -2.6-2.4-2.2-2 0 0.2 0.4 0.6 0.8 1 1.2 1.4B + , 1.5T c V r e (r) / T c rT c a τ =1/38T c a τ =1/45T c FIG. 11: The results for V o re ( ~r ; T ) extracted from lattices with different discretization levels. Shown are the results at 1.2 T c (top panels) and 1.5 T c (bottom panels). The left panels correspond to the L=0 channel while the right panels show the L=1channel results. we added a small overall additive constant to the results at the coarser lattice. We see that the discretization erroris much smaller compared to the other uncertainties in our calculation. We therefore take the results from our finestset, set 3, as indicative of the continuum results. These are the results shown in Sec. IV A. 2. Systematics for V o im ( ~r ; T ) For V o im ( ~r ; T ) the smearing dependence of the extracted results is shown in Figure 12. For the B z operator, theresults stabilize quickly: for set 3, already by 200 smearing steps the results seem to have stabilized. For B + it issimilar, except at 1.2 T c at longer distances. Like for V o re ( ~r ; T ), the error bands quoted in Sec. IV B include the spreadbetween smearing levels 300 and 400 for set 3, and that between smearing levels 200 and 250 for set 2.In comparison to the large errors associated with the extraction of V o im ( ~r ; T ), the cutoff effects do not seem to besignificant. In Figure 13 we compare the results for V o im ( ~r ; T ) from Set 1 and set 2. Given the small cutoff dependence,we treat the results from our finest lattices as indicative of continuum results. These are the results shown in Sec.IV B. [1] T. Matsui and H. Satz, Phys. Lett. B 178 (1986) 416.[2] A. Rothkopf, Phys. Rept. 858 (2020) 1.[3] For compact reviews, with also discussion of phenomenology, seeA. Mocsy, P. Petreczky & M. Strickland, Int. J. Mod. Phys. A 28 (2013) 1340012.S. Datta, Pramana 84 (2015) 881.[4] M. Laine, O. Philipsen, P. Romatschke & M. Tassler, J. H. E. P. Phys. Rev. D 77 (2008) 014017.[6] Y. Akamatsu, Phys. Rev. D 87 (2013) 045016.[7] Y. Akamatsu, Phys. Rev. D 91 (2015) 056002.J.P. Blaizot, D. De Boni, P. Faccioli & G. Garberoglio, Nucl. Phys. A 946 (2016) 49.N. Brambilla, M. A. Escobedo, J. soto & A. Vairo, Phys. Rev. D 97 (2018) 074009.R. Sharma & A. Tiwari, Phys. Rev. D 101 (2020) 074004. z c a τ V i m (r) r/a s + c a τ V i m (r) r/a s z c a τ V i m (r) r/a s + c a τ V i m (r) r/a s FIG. 12: Smearing dependence of the imaginary part of the potential extracted from the hybrid operator inserted Wilson loops.The top row shows results at 1.2 T c while the bottom row shows results at 1.5 T c . The panels to the left are for B z and thoseto the right are for B + insertions. For viewing purpose some sets have been slightly shifted along x axis in the plot. z , 1.2T c V i m (r) / T c rT c a τ =1/38T c a τ =1/45T c + , 1.2T c V i m (r) / T c rT c a τ =1/38T c a τ =1/45T c z , 1.5T c V i m (r) / T c rT c a τ =1/38T c a τ =1/45T c + , 1.5T c V i m (r) / T c rT c a τ =1/38T c a τ =1/45T c FIG. 13: Comparison of results for V o im ( ~r ; T ) at two different lattice spacings. The top row shows results at 1.2 T c while thebottom row shows results at 1.5 T c . The panels to the left are for B z and those to the right are for B + insertions. [8] A. Rothkopf, T. Hatsuda & S. Sasaki, Phys. Rev. Lett. 108 (2012) 162001.[9] Y. Burnier, O. Kaczmarek & A. Rothkopf, Phys. Rev. Lett. 114 (2015) 082001.Y. Burnier & A. Rothkopf, Phys. Rev. D 95 (2017) 054511.[10] P. Petreczky & J. weber, Nucl. Phys. A 967 (2017) 592.[11] D. Bala & S. Datta, Phys. Rev. D 101 (2020) 034507.[12] R. Sharma & I. Vitev, Phys. Rev. C 87 (2013) 044905.[13] Y. Akamatsu & A. Rothkopf, Phys. Rev. D 85 (2012) 105011.[14] S. Kajimoto, Y. Akamatsu, M. Asakawa & A. Rothkopf, Phys. Rev. D 97 (2018) 014003.[15] O. Philipsen & M. Wagner, Phys. Rev. D 89 (2014) 014509.[16] K.J. Juge, J. Kuti & C. Morningstar, Phys. Rev. Lett. 90, 161601 (2003).S. Capitani, O. Philipsen, C. Reisinger, C. Riehl & M. Wagner, Phys. Rev. D 99 (2019) 034502[17] N. Brambilla, A. Pineda, J. Soto & A. Vairo, Nucl. Phys. B 566 (2000) 275.[18] G. Bali & A. Pineda, Phys. Rev. D 69 (2004) 094001.[19] F. Zantow, O. Kaczmarek, F. Karsch & P. Petreczky, Proceedings, SEWM 2002 (hep-lat/0301015).[20] A. Bazavov, N. Brambilla, P. Petreczky, A. Vairo and J.H. Weber, Phys. Rev. D 98 (2018) 054511.[21] D. Bala & S. Datta, Proceedings, Lattice 2019 (arxiv:1912.04826).[22] P. Petreczky, A. Rothkopf & J. weber, Nucl. Phys. A 982 (2019) 735.[23] Y. Burnier & A. Rothkopf, Phys. Rev. D 87 (2013) 114019.[24] T. R. Klassen, Nucl. Phys. B 533 (1998) 557.[25] M. L¨uscher & P. Weisz, J. H. E. P. Phys. Lett.